In the first moments of the TV comedy “Catastrophe,” an American man has a fling with an Irish woman in his hotel room in London. What follows is a sudden plunge into cross-cultural courtship, pregnancy and marriage. In a debut season of just six half-hour episodes, “Catastrophe” covers about eight months in the uncertain (and often off-color) lives of its lead characters.

The life of the show itself has been full of whiplash for creators and stars Rob Delaney and Sharon Horgan. “Catastrophe” first premiered in the U.K. last January to rave reviews and a barrage of social-media response each week when it aired on public broadcaster Channel 4. Since “Catastrophe” appeared in the U.S. last month on Amazon Prime, the reaction has been more muted. Though some U.S. critics weighed in early, and Amazon tried to maximize exposure by making the first episode freely available on Facebook, viewers tend to discover streaming TV shows on their own and digest them on staggered schedules.

“When it launched on Amazon I was expecting that same kind of crazy influx of commentary [as in Britain], but it was nothing like that,” says Ms. Horgan, who has produced and starred in several previous U.K. TV shows. “Instead it’s been drip, drip, drip, all the way. I was like, ‘Oh, sh—, is it failing?’”

Quite the opposite. “Catastrophe” is one of TV’s sleeper hits of the summer, judging from websites that tabulate scores based on critics’ reviews. On Metacritic, “Catastrophe” ranks among the top 10 TV series of the last three months, and on Rotten Tomatoes it is No. 1 among new shows. Amazon doesn’t share viewership data, but in its quest to build a stable of quality original series for grown-ups, the company has clearly found a companion to its Golden-Globe-winning series “Transparent.”

“The superlatives are scary and nice, but I am concerned that we are being tricked,” Mr. Delaney says.

He’s a comedian who established his career by telling jokes on Twitter (he has 1.2 million followers), and that’s where he first struck up a friendship with Ms. Horgan. The show they created avoids saccharine rom-com conventions with characters who talk baldly about sex, the indignities of pregnancy and their mixed feelings about each other. The key to the fictional couple’s relationship—and the charm of “Catastrophe”—is the fact that they’re good at making each other laugh.

In a few weeks, the producers will start shooting season 2, which got a green light before “Catastrophe” even hit the air in the U.K. Having left off with Rob and Sharon’s wedding and a pregnancy-related cliffhanger, the story will jump “a few years into the future,” says Mr. Delaney, a Boston native who moved to London with his family to produce the show.

The series is expected to air again in the fall with another installment of six episodes. That length is standard in the U.K., but might seem like a half measure to U.S. viewers accustomed to TV seasons of up to 24 episodes each.

Mr. Delaney says, “For all the Americans that will make angry, they will have gotten a total of 12 episodes within a year. So I don’t want to hear anybody whining.”

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